Parametric Resonance Applications in Neutrophil Dynamics

In the treatment of malignant tumors, chemotherapy is frequently accompanied by hematopoietic side effects due to the myelosuppressive character of drugs used. A common neutropenia(accompanied by fever and possible infection), to a lesser extent, thrombocytopenia and anemia are frequently observed. To circumvent these side effects, an effective administration of granulocyte colony stimulating factor (G-CSF) is suggested in clinic practice [1-7], and now a standard postchemotherapy treatment for neutropenia. Mathematically, people try to understand G-CSF control by a variety of models [3,8-12].


Introduction
In the treatment of malignant tumors, chemotherapy is frequently accompanied by hematopoietic side effects due to the myelosuppressive character of drugs used. A common neutropenia(accompanied by fever and possible infection), to a lesser extent, thrombocytopenia and anemia are frequently observed. To circumvent these side effects, an effective administration of granulocyte colony stimulating factor (G-CSF) is suggested in clinic practice [1][2][3][4][5][6][7], and now a standard postchemotherapy treatment for neutropenia. Mathematically, people try to understand G-CSF control by a variety of models [3,[8][9][10][11][12].
A significant G-CSF support in clinic perspective is periodically repeated treatment of chemotherapy, and neutrophil is highly responsive and dependent on timing and protocol of the drug's administration [13,14]. Experiment have reported that chemotherapy increases apoptosis in both proliferative HSCs (hematopoietic stem cells) and proliferative neutrophil precursors [15]. To idealize the effects of chemotherapy and G-CSF, a square wave temporal functions for the loss rates,neutrophil precursor proliferative rate are used, explained in reference [16]. The resonance between the perturbation due to the periodic chemotherapy and the intrinsic characteristic frequency in the neutrophil production have been discussed. The periodic damped oscillation in neutrophil levels [17] are also reported with the assumption of single dose of chemotherapy, which is simulated by exponential mathematically.
In this paper, we explain the hematopoietic dynamics with the pharmacokinetics of chemotherapy and G-CSF by periodic parametric excitation. The dependence of the neutrophil response on the stimulation frequency is examined and resonance phenomena is discussed.

The Model Equations
The combined dynamics of hematopoietic stem cells and neutrophil production is illustrated in Figure 1 and generally described by the following equations where Holling function The neutrophil system (1) consists of DDEs(delay differential equations) with two delays. These two equations takes into account the balance between the net production and loss terms of HSCs and circulating neutrophils, where Q and N (both with unit cells/kg)are respectively hematopoietic stem cells and neutrophils. HSCs differentiate into the committed neutrophil compartment with term k N (N)Q. Delay τ s denotes their duration days of proliferative phase of HSCs. HSCs either remain in the resting phase or enter into the proliferative phase at a rate β(Q). The differentiation of HSCs to neutrophil is controlled by k N (N) (the differentiated rate of circulating neutrophils), and HSCs proliferation is controlled by the resting HSCs population with proliferation rate β(Q). r s represents the apoptosis rate during the proliferation phase of HSCs. The circulating neutrophils survive with their averaged lifespan 1 N γ − and thus with the ature death N N γ . Cells in the neutrophil lines experience a successive division with time period τ NP and then enter into a purely maturation compartment with time period τ NM , and the total time period τ N =τ NP +τ NM .η NP denotes the proliferative rate of neutrophils and γ o denotes their death rate before neutrophils enter into the circulation. The parameter values in System (1).
Suppose chemotherapy is administrated periodically. Neutropenia is a common side-effect, and G-SCF is also administrated after chemotherapy. Both the effects of chemotherapy and G-SCF are maintained for several days. Chemotherapy increases apoptosis in both proliferative HSCs and proliferative neutrophil precursors. The perturbation of parameter values are supposed to be

Resonance
The response of neutrophil to chemotherapy, and the effect of G-CSF adminstration always can be strong as varying excitation frequency near natural frequency. Under the excited vibration of parameters r s ,η NP and γ o , resonance oscillation can be initiated in the interval of the excitation frequency due to the nonlinearity. It is well known, system can be unstable due to the action of parametric excitation. In certain intervals of frequency Ω lying,system(1) may have phase lock solutions. Set ω 1 = ω 2 = ω 3 =Ω, and then let = p q ω Ω , in general, there exist intervals of excitation frequency Ω with bigger width to induce 1:1 primary resonance or 1:2 subharmonic resonance phenomena. The combination parametric resonance can be triggered which induce the periodic solutions with large amplitude. To perform bifurcation analysis, we choose Ω as bifurcation parameter, then define µ:=( Ω,τ s ), and set .φ µ (Q,N;t) to be solution flow of system (1) with initial values Q(t-τ)= Q * ,N(t-τ)=N * for 0 t τ ≤ ≤ , where τ=max(τ s, τ N ). Since Ω:ω=1:1, every point on its orbit is a period point. Define the amplitude || N|| to describe the difference between the maximal period point and the minimal period point. We varied Ω widely and, for each value, solved DDEs (1) for t=1200 days using initial values Q= Q * ,N=N * for 0 N t t τ − ≤ ≤ due to delays τ s <τ N satisfied. The long term effect of chemotherapy and G-CSF for each simulation the last 400 days are used to obtain the amplitude in neutrophil fluctuations. || N|| continue to a bigger value to form the resonance in a widely varying scope of excitation frequency. The 1:1 resonance regime in parameter space (Ω,τ s )plane is formed in a shaded regime as shown in Figure 2a.
In special, choose τ s =3.2, the response of neutrophil numbers near 1 (the ratio of Ω/ω) are denoted by its amplitude exhibits large values continuously. The amplitude response corresponding to the ratio of Ω/ω is drawn in Figure 2b.
Vary excitation frequency Ω near ω/2 with ω being natural frequency, the 1:2 subharmonic resonance regime in parameter space (,τ) plane is also formed by shaded regime in Figure 3a. In special, choose τ s =3.2, the response of neutrophil numbers near 0.5 (the ratio of Ω/ω) are drawn which exhibits 1:2 subharmonic resonance phenomena as shown in Figure 3b. The amplitude response exhibits large values continuously.

Conclusion
In a physiologically realistic point view, a two compartment model combined the dynamics of hematopoietic stem cells and the differentiated neutrophils with its response dependence on the period chemotherapy and G-CSF adminstration are given. The combined parametric excitation resonance phenomena are discussed. When the ratio between excitation frequency and eigenfrequency are rational, both 1:1 primary resonance and 1:2 subharmonic resonance are tracked which respectively form a resonance regime on plane of parameters.